An introduction to finite volumes for gas dynamics

TL;DR

This paper provides an accessible introduction to finite volume methods for gas dynamics, covering fundamental concepts, extensions to systems, and boundary condition treatments, suitable for learners and practitioners.

Contribution

It offers a clear, elementary presentation of finite volume techniques applied to gas dynamics, including extensions to systems and multi-dimensional cases.

Findings

Effective upwind scheme implementation for 1D gas dynamics

Extension of methods to hyperbolic systems and 2D cases

Detailed treatment of boundary conditions in finite volume methods

Abstract

We propose an elementary introduction to the finite volume method in the context of gas dynamics conservation laws. Our approach is founded on the advection equation, the exact integration of the associated Cauchy problem, and the so-called upwind scheme in one space dimension. It is then extended in three directions: hyperbolic linear systems and particularily the system of acoustics, gas dynamics with the help of the Roe matrix and two space dimensions by following the approach proposed by Van Leer. A special emphasis on boundary conditions is proposed all along the text.